Abstract
This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by mathcal {U}_N. We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from mathcal {U}_N onto some open convex subset of {mathbb {R}}^{2N} and allow to solve the equation by quadrature for any such initial datum. As a consequence, mathcal {U}_N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. https://doi.org/10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on mathcal {U}_N is given by a polynomial characterization and a spectral characterization of the manifold mathcal {U}_N. Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.
Highlights
The Benjamin–Ono (BO) equation on the line reads as∂t u = H∂x2u − ∂x (u2), (t, x) ∈ R × R, (1.1)where u is real-valued and H = −isign(D) : L2(R) → L2(R) denotes the Hilbert transform, D = −i∂x,H f (ξ ) = −isign(ξ ) f(ξ ), ∀ f ∈ L2(R). (1.2)sign(±ξ ) = ±1, for all ξ > 0 and sign(0) = 0, f ∈ L2(R) denotes the Fourier– Plancherel transform of f ∈ L2(R)
The goal of this paper is to show the complete integrability of equation (1.1) when restricted to every multi-soliton manifold
We show that ω establishes the symplectic structure, which corresponds to the Gardner bracket (1.5), on the N -soliton manifold UN defined by (1.8)
Summary
Proposition 1.9 is restated with more details in Theorem 4.8, Proposition 5.4 and Corollary 5.5, which both give a spectral characterization of the N -soliton manifold UN and establish a spectral connection between the inverse spectral matrix M(u) ∈ CN×N and the Lax operator Lu, which is given in Definition 2.2, of the BO equation (1.1), for any u ∈ UN. In the space non-periodic regime, we do not know whether there exists a large submanifold of L2(R, R), denoted by L, such that L contains every multi-soliton manifold UN , L is invariant under the flow of (1.1), and there exist action–angle coordinates for the BO equation (1.1) on L, whose restriction to UN is N given in (1.14). We give a geometric description of the inverse spectral transform by proving the real bi-analyticity and the symplectomorphism property of the action–angle map given by (1.14). We refer to Sun [20,21] to see the long time and asymptotic behavior of other NLS–Szegoequations
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